LAS VEGAS - A confident Mitt Romney criticized his Republican rivals and fended off attacks on immigration and healthcare Tuesday at a debate that could help reinstall him as the party's presidential front-runner.

Romney, the former Massachusetts governor, overshadowed businessman Herman Cain and Texas Governor Rick Perry, his two main challengers for the nomination to take on President Barack Obama in 2012.

Seeking to convince skeptical conservatives to get behind him, Romney fought off attacks from Perry, who brought up an old charge that he hired illegal immigrants to cut his lawn.

After a testy exchange, Romney hit back.

"Texas has had a 60 percent increase in illegal immigrants in Texas. If there's someone who has a record as governor with regards to illegal immigration that doesn't stand up to muster, it's you, not me,'' he said.

Polls show Romney has the best chance of any Republican of defeating Obama, whose approval ratings have dropped as he struggles to revive the economy and cut the unemployment rate from 9 percent.

But pizza magnate Cain has headed recent polls of Republicans, many of whom think Romney is not conservative enough to deserve the nomination.

Romney, a far more confident and polished debater than in his 2008 presidential run, put Perry on the defensive over an evangelical pastor with ties to the Texas governor who criticized Mormonism.

"The idea that we should choose people, based on their religion, for public office is what I find to be most troubling,'' said Romney, a Mormon. Perry replied that he did not agree with what the preacher had said.

The two men pointed fingers and shouted over each other as Perry brought up a story from Romney's 2008 campaign, that he had hired a contractor that employed illegal Latin American workers to work on Romney's lawn. Romney fired the company when he found out about the undocumented workers.

"Mitt, you lose all of your standing, from my perspective, because you hired illegals in your home and you knew about it for a year,'' said Perry.

Cain under attack

Cain was a central target at the lively CNN-sponsored debate where candidates criticized his 9-9-9 tax reform plan.

"Herman, I love you brother, but let me tell you something. You don't need to have a big analysis to figure this out,'' said Perry, who showed a more robust performance than in recent debates.

Cain, the former Godfather's Pizza CEO who has shot to the top of Republican opinion polls in the last few weeks, found himself taken seriously by rivals who had previously believed he would simply fade as time wore on.

Cain insisted his plan to reduce personal income and corporate taxes to 9 percent and create a 9 percent national sales tax would not raise taxes on middle-income Americans despite expert analysis that it would.

But instead of specifically explaining how his plan would bring in more revenue than the current tax system and why middle Americans would not pay higher taxes, he simply said his rivals were wrong and that they should read through his plan on his campaign website.

"That simply is not true,'' he said at one point. "I invite people to look at our analysis which we make available.''

The flaw cited by the candidates in Cain's plan is the 9 percent national sales tax, which would hit all Americans who already pay state sales taxes. A U.S. national sales tax does not exist and Republicans are wary of any effort to craft a European-style value-added tax.

Romney's rivals again brought up the healthcare plan Romney fathered as governor of Massachusetts that Democrat Obama has said was a model for the 2010 overhaul the White House engineered, which conservatives want to repeal.

The Massachusetts plan is perhaps Romney's biggest Achilles heel in persuading conservatives he is one of them.

"You just don't have credibility, Mitt, when it comes to repealing Obamacare,'' said long-shot candidate Rick Santorum. ''Your plan was the basis for Obamacare.''

When former House of Representatives Newt Gingrich chimed in on the point, Romney told him a main feature of the Massachusetts plan, an individual mandate, had actually been Gingrich's idea.

"Actually, Newt, we got the idea of an individual mandate from you,'' he said.

"Wait a second,'' said Gingrich. "What you said is not true.''